Ex 12.3, 10
The area of an equilateral triangle ABC is 17320.5 cm2 . With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see figure). Find the area of the shaded region. (Use = 3.14 and 3= 1.73205)
Area of shaded region
= Area of equilateral triangle
Area of sector ADE
Area of sector BDF
Area of sector CFE
It is given that,
ABC is an equilateral triangle,
So, sides of an triangle = AB = BC = AC
Also,
Area of an equilateral triangle = 17320.5 cm2
3/4 (side)2 = 17320.5
(Side)2 = (17320.5 4)/ 3
(Side)2 = (17320.5 4)/1.73205
(Side)2 = (17320.5 4 100000)/173205
(Side)2 = (173205 4 10000)/173205
(Side)2 = 40000
Side = 40000
Side = (4 10000)
Side = ((2)^2 (100)^2 )
Side = 200 cm
Given
Radius of each circle = Half length of side of triangle
= 200/2
= 100 cm
We know that equilateral triangle has all angles 60
ABC = BAC = ACB = 60
So, = 60
Area of sector ADE = /(360 ) 2
= (60 )/360 3.14 (100)2
= 1/6 3.14 100 100
= 1/3 3.14 50 100
= 15700/3 cm2
Area of sector DBF & CFE have the same value of radius and angle
Hence ,
Area sector ADE = Area sector DBF = Area sector CFE = 15700/3 cm2
Area of shaded region
= Area of equilateral triangle
Area of sector ADE
Area of sector BDF
Area of sector CFE
= 17320.5 15700/3 15700/3 15700/3
= 17320.5 3 15700/3
= 17320.5 15700
= 1620.5 cm2
Hence, area of shaded region = 1620.5 cm2

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.